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Question

Difficulty: HardIntegers, Absolute Value, and Number Lines

On a standard number line, the coordinate of point PP is an integer pp, and the coordinate of point QQ is an integer qq. The distance between PP and the origin is less than 5, and the distance between QQ and −3-3 is exactly 4. If the product p⋅qp \cdot q is minimized, what is the value of ∣p−q∣|p - q|?

  1. A
    3
  2. B
    5
  3. C
    8
  4. 11Answer
  5. E
    12

Answer

The value of ∣p−q∣|p - q| is 11, which corresponds to the option with a value of 11.
The distance between PP and the origin is less than 5, so the integer coordinate pp must satisfy ∣p∣<5|p| < 5, meaning p∈{−4,−3,−2,−1,0,1,2,3,4}p \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}. The distance between QQ and −3-3 is exactly 4, so ∣q−(−3)∣=4  ⟹  ∣q+3∣=4|q - (-3)| = 4 \implies |q + 3| = 4, which gives q=1q = 1 or q=−7q = -7. To minimize the product p⋅qp \cdot q, we analyze the two possibilities for qq. If q=1q = 1, the minimum product is −4-4 when p=−4p = -4. If q=−7q = -7, the minimum product is −28-28 when p=4p = 4. The absolute minimum product is −28-28, achieved when p=4p = 4 and q=−7q = -7. The value of ∣p−q∣|p - q| for this pair is ∣4−(−7)∣=11|4 - (-7)| = 11.

Step-by-Step Solution

1
Determine the possible integer coordinates for point PP.
p∈{−4,−3,−2,−1,0,1,2,3,4}p \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}
The distance from PP to the origin is less than 5, meaning ∣p∣<5|p| < 5. Since pp is an integer, it can be any integer strictly between −5-5 and 55.
2
Determine the possible integer coordinates for point QQ.
q=1q = 1 or q=−7q = -7
The distance from QQ to −3-3 is exactly 4, meaning ∣q−(−3)∣=4  ⟹  ∣q+3∣=4|q - (-3)| = 4 \implies |q + 3| = 4. Solving this gives q+3=4  ⟹  q=1q + 3 = 4 \implies q = 1, or q+3=−4  ⟹  q=−7q + 3 = -4 \implies q = -7.
3
Evaluate products of pp and qq to find the pair (p,q)(p, q) that minimizes p⋅qp \cdot q.
p=4p = 4 and q=−7q = -7, yielding the minimum product of −28-28.
If q=1q = 1, the minimum product is p⋅1=−4p \cdot 1 = -4 when p=−4p = -4. If q=−7q = -7, the product is −7p-7p; to minimize this negative product, we choose the largest positive value for pp, which is 44, yielding −28-28. Comparing −4-4 and −28-28, the absolute minimum is −28-28.
4
Compute the absolute difference ∣p−q∣|p - q| for the minimizing pair.
∣4−(−7)∣=11|4 - (-7)| = 11
Using the coordinates p=4p = 4 and q=−7q = -7, we find the distance between them on the number line.

Key Concept

Distance on a number line can be calculated using absolute value. Finding the minimum of a product involving signed integers requires evaluating both positive and negative cases.
Estimated Time:2m 0s
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Topics

Integers, Absolute Value, and Number Lines

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